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31\. Mislabeled seafood In 2013 the environmental group Oceana (usa.oceana.org) analyzed 1215 samples of seafood purchased across the United States and genetically compared the pieces to standard gene fragments that can identify the species. Laboratory results indicated that \(33 \%\) of the seafood was mislabeled according to U.S. Food and Drug Administration guidelines. a. Construct a \(95 \%\) confidence interval for the proportion of all seafood sold in the United States that is mislabeled or misidentified. b. Explain what your confidence interval says about seafood sold in the United States. c. A 2009 report by the Government Accountability Office says that the Food and Drug Administration has spent very little time recently looking for seafood fraud. Suppose an official said, "That's only 1215 packages out of the billions of pieces of seafood sold in a year. With the small number tested, I don't know that one would want to change one's buying habits." (An official was quoted similarly in a different but similar context). Is this argument valid? Explain.

Step by step solution

01

Calculate Point Estimate

The point estimate of the proportion of all seafood sold in the United States that is mislabeled or misidentified is given by the proportion in the sample, which is \(0.33\)

02

Determine Level of Confidence

The problem asks for a \(95\%\) confidence interval, which corresponds to a Z-value of \(1.96\) (found in standard statistical tables)

03

Apply Confidence Interval Formula

The formula for the confidence interval for a proportion is \(p \pm z\sqrt{((p(1-p))/n)}\), where \(p\) is the point estimate, \(z\) is the Z-score corresponding to the desired level of confidence, and \(n\) is the sample size. Plugging into this formula, the confidence interval will be \(0.33 \pm 1.96\sqrt{((0.33(1-0.33))/1215)}\)

04

Calculate and Interpret Confidence Interval

Based on the formula in step 3, you can solve for the confidence interval, which represents the range within which we can be \(95\%\) confident that the actual proportion of mislabeled seafood lies in the population.

05

Evaluate Statistical Argument

By comparing the confidence interval to any assumptions or statements, you assess the validity of the argument. The consequence of the small sample size is indicated by the width of the confidence interval. If the confidence interval is wide, it implies considerable uncertainty and might lend some support to the official's argument.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mislabeled Seafood

Mislabeled seafood refers to a situation where the seafood sold is incorrectly labeled as a different species. This is a critical issue as it can mislead consumers and potentially involve health or ethical concerns. When examining the problem of mislabeled seafood, it's essential to investigate factors such as regulatory adherence and consumer trust. According to the study by Oceana, approximately 33% of the seafood sampled in the United States was found to be mislabeled according to FDA guidelines. This proportion provides a foundation for statistical analysis and confidence interval calculation, helping us understand the broader scope of the issue in the overall market. Mislabeling can affect prices, as higher-value seafood might be substituted with a less expensive alternative. So, addressing seafood mislabeling is crucial for maintaining consumer protection and market fairness.

Proportion Estimation

Proportion estimation helps us determine the likelihood of a certain characteristic within a population, based on a sample. In this context, proportion estimation was used to identify the percentage of mislabeled seafood. The calculation begins with finding the point estimate, which is done by using the proportion found in the sample, here which was 33% or 0.33. This means that 33 out of 100 pieces of seafood in the sample were found to be mislabeled.

• Deployment of proportion estimation is crucial in drawing conclusions about larger populations.
• Enables the creation of a confidence interval, showing the expected range of the true proportion in the broader market.

Thus, such estimations afford a statistical portrayal of the true magnitude of mislabeling in a population, despite never inspecting every single item sold.

Z-Score

The Z-score plays a vital role in the calculation of confidence intervals. It is a statistic that measures the number of standard deviations a data point is from the mean, allowing for the identification of how typical or unusual a sample's statistic is within a wider population. For a confidence interval of 95%, the corresponding Z-score is 1.96.

• Z-scores offer a standardized method to determine confidence levels.
• For a 95% confidence interval, this describes a range covering roughly 95% of all observed data values.

Incorporating the Z-score into our formula helps ascertain the degree of certainty with which we estimate the actual proportion of mislabeled seafood in the general market. This number essentially dictates the breadth of the confidence interval, reflecting the reliability of our estimate.

Sample Size

Sample size is foundational in statistical analysis and plays a crucial part in forming confidence intervals. It represents the number of observations or data points included in the sample. In the case of the mislabeled seafood study, 1215 samples were taken. The larger the sample size, the more accurate and narrow the confidence interval could potentially become.

• A larger sample size tends to provide more accurate representation of the population.
• Reduces the margin of error, thus leading to narrower confidence intervals.

This is why the statement by the official that the sample size is too small should be scrutinized. While 1215 samples may seem small compared to the billions sold, it's statistically substantial enough to provide a reliable estimation, as long as the sampling was done correctly and unbiasedly.